σ-asymptotically lacunary statistical equivalent sequences

• Autorzy: Ekrem Savaş , Richard Patterson
• Języki publikacji: EN

Abstrakty

EN

This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 $$\mathop {\lim }\limits_r \frac{1}{{h_r }}\left\{ {k \in I_r :\left| {\frac{{x_{\sigma ^k (m)} }}{{y_{\sigma ^k (m)} }} - L} \right| \geqslant \in } \right\} = 0$$ uniformly in m = 1, 2, 3, ..., (denoted by x $$\mathop \sim \limits^{S_{\sigma ,\theta } } $$ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented.

Słowa kluczowe

EN

σ-asymptotically; Equivalent of multiple L; 40A99; 40A05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 4
• Strony: 648-655

Daty

wydano

2006-12-01

online

2006-12-01

Twórcy

autor

Ekrem Savaş

• Yüzüncü Yil University

autor

Richard Patterson

• University of North Florida

Bibliografia

• [1] G. Das and B.K. Patel: “Lacunary distribution of sequences”, Indian J. Pure Appl. Math., Vol. 26(1), (1989), pp. 54–74.
• [2] H. Fast: “Sur la convergence statistique”, Collog. Math., Vol. 2, (1951), pp. 241–244.
• [3] J.A. Fridy: “Minimal rates of summability”, Can. J. Math., Vol. 30(4), (1978), pp. 808–816.
• [4] J.A. Fridy: “On statistical sonvergence”, Analysis, Vol. 5, (1985), pp. 301–313.
• [5] J.A. Fridy and C. Orhan: “Lacunary statistical sonvergent”, Pacific J. Math., Vol. 160(1), (1993), pp. 43–51.
• [6] G.G. Lorentz: “A contribution to the theory of divergent sequences”, Acta. Math., Vol. 80, (1948), pp. 167–190. http://dx.doi.org/10.1007/BF02393648
• [7] Mursaleen: “Some new spaces of lacunary sequences and invariant means”, Ital. J. Pure Appl. Math., Vol. 11, (2002), pp. 175–181.
• [8] Mursaleen: “New invariant matrix methods of summability”, Quart. J. Math. Oxford, Vol. 34(2), (1983), pp. 133, 77–86.
• [9] M. Marouf: “Asymptotic equivalence and summability”, Int. J. Math. Math. Sci., Vol. 16(4), (1993), pp. 755–762. http://dx.doi.org/10.1155/S0161171293000948
• [10] R.F. Patterson: “On asymptotically statistically equivalent sequences”, Demonstratio Math., Vol. 36(1), (2003), pp. 149–153.
• [11] R.F. Patterson and E. Savaş: “On asymptotically lacunary statistically equivalent sequences”, (in press).
• [12] P. Schaefer: “Infinite matrices and invariant means”, Proc. Amer. Math. Soc., Vol. 36, (1972), pp. 104–110. http://dx.doi.org/10.2307/2039044

Identyfikatory

DOI

10.2478/s11533-006-0031-8